Does “Air Launch to Orbit” makes the best use of Earth's Rotation?

Usually, all artificial satellites are launched in the prograde direction, since this minimizes the amount of propellant required to reach orbit by taking advantage of the Earth's rotation. Further, by launching from the equator, we can make use of Earth's rotation to the maximum level.

Air launch to orbit is the method of launching rockets from an aircraft after it reaches a certain altitude and direction. This method has some advantages over launching from a fixed launchpad on the ground.

Is launching a rocket from an aircraft along the rotation of the Earth (prograde) beneficial or it doesn't matter? In other words, does air launch to orbit makes the best use of launching in the prograde direction over retrograde direction? If yes, why are we not launching from the Equator to obtain the maximum benefit of Earth's rotation? If the aircraft takes off from a non-equatorial runway, does it gain any additional speed when it travels to the equator? Is the answer dependent on the local wind stream around the carrier-aircraft?

For the sake of reference, the following image shows the Flight Plan for a Pegasus Rocket Launch from the L-1011 Stargazer Aircraft:

Image Source : SpaceFlight101.com

It can be seen the launch (shown by $$\mathrm{\color{magenta}{magenta}}$$ coloured line) is towards East (not exactly East) or it's prograde.

• Your question is attracting close votes for being unclear. For example, why are prograde/retrograde advantages different for air launch regarding where the launcher is in respect to the equator? How about for sea launch? Consider an edit to your question to make it clear what you are actually asking. And be careful of assumptions - air launch has many drawbacks over conventional pads. – Organic Marble Nov 3 at 15:06
• When an aircraft lifts off, the rotational speed it originally had from the Earth doesn't magically vanish. – Organic Marble Nov 3 at 15:50
• @Hobbes, Could you please explain, how can the answers to the mentioned question answer my question? – Intellex Nov 3 at 16:02
• @Intellex Nations generally launch from as close to the equator as they can. Many don't have equatorial territories or have other constraints. Mobile launch pads (air or sea) have been used but are extremely constraining on the size of the launch vehicles they can support as well as having many other disadvantages. – Organic Marble Nov 3 at 16:05
• Air launch or ground launch makes no difference: the atmosphere co-rotates with the surface plus or minus the wind speed. As Earth's rotation at the equator is much faster than the highest wind speeds, the linked question gives a good approximation of how beneficial a prograde launch is. – Hobbes Nov 3 at 17:59

To launch Pegasus from the equator, the L-1011 would have to take off from Florida, fly 3000 km south, launch, then fly 3000 km home -- pretty close to the range limit for that aircraft. At US\$2 per gallon, that's about \$50,000 in fuel.